§21.6 Products

Let $\mathbf{T}=[T_{jk}]$ be an arbitrary $h\times h$ orthogonal matrix (that is, $\mathbf{T}{\mathbf{T}}^{\mathrm{T}}=\mathbf{I}$) with rational elements. Also, let $\mathbf{Z}$ be an arbitrary $g\times h$ matrix. Define

21.6.1		$$𝒦={\mathbb{Z}}^{g\times h}\mathbf{T}/({\mathbb{Z}}^{g\times h}\mathbf{T}\cap {\mathbb{Z}}^{g\times h}),$$
ⓘ Defines: $𝒦$: set of matrices (locally) Symbols: $\mathbb{Z}$: set of all integers, $\cap$: intersection, $g$: positive integer and $h$: positive integer Permalink: http://dlmf.nist.gov/21.6.E1 Encodings: TeX, pMML, png See also: Annotations for §21.6(i), §21.6 and Ch.21

that is, $𝒦$ is the set of all $g\times h$ matrices that are obtained by premultiplying $\mathbf{T}$ by any $g\times h$ matrix with integer elements; two such matrices in $𝒦$ are considered equivalent if their difference is a matrix with integer elements. Also, let

21.6.2		$$𝒟=|{\mathbf{T}}^{\mathrm{T}}{\mathbb{Z}}^h/({\mathbf{T}}^{\mathrm{T}}{\mathbb{Z}}^h\cap {\mathbb{Z}}^h)|,$$
ⓘ Defines: $𝒟$: number of elements (locally) Symbols: $\mathbb{Z}$: set of all integers, $\cap$: intersection and $h$: positive integer Permalink: http://dlmf.nist.gov/21.6.E2 Encodings: TeX, pMML, png See also: Annotations for §21.6(i), §21.6 and Ch.21

that is, $𝒟$ is the number of elements in the set containing all $h$-dimensional vectors obtained by multiplying ${\mathbf{T}}^{\mathrm{T}}$ on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. Then

21.6.3		$$\begin{array}{l}\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{\mathbf{z}}_k\right|\mathbf{\Omega })\\ =\frac{1}{{𝒟}^g}\sum _{\mathbf{A}\in 𝒦}\sum _{\mathbf{B}\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}tr\left[\frac{1}{2}{\mathbf{A}}^{\mathrm{T}}\mathbf{\Omega }\mathbf{A}+{\mathbf{A}}^{\mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\prod _{j=1}^h\theta \left({\mathbf{z}}_j+\mathbf{\Omega }{\mathbf{a}}_j+{\mathbf{b}}_j\right|\mathbf{\Omega }),\end{array}$$
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $tr\mathbf{A}$: trace of matrix $\mathbf{A}$, $g$: positive integer, $h$: positive integer, $\mathbf{\Omega }$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements Permalink: http://dlmf.nist.gov/21.6.E3 Encodings: TeX, pMML, png See also: Annotations for §21.6(i), §21.6 and Ch.21

where ${\mathbf{z}}_j$, ${\mathbf{a}}_j$, ${\mathbf{b}}_j$ denote respectively the $j$th columns of $\mathbf{Z}$, $\mathbf{A}$, $\mathbf{B}$. This is the Riemann identity. On using theta functions with characteristics, it becomes

21.6.4		$$\begin{array}{l}\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{\sum }_{k=1}^h{T}_{jk}{\mathbf{c}}_k}{{\sum }_{k=1}^h{T}_{jk}{\mathbf{d}}_k}\right]\left(\sum _{k=1}^h{T}_{jk}{\mathbf{z}}_k\right|\mathbf{\Omega })\\ =\frac{1}{{𝒟}^g}\sum _{\mathbf{A}\in 𝒦}\sum _{\mathbf{B}\in 𝒦}{\mathrm{e}}^{-2\pi \mathrm{i}{\sum }_{j=1}^h{\mathbf{b}}_j\cdot {\mathbf{c}}_j}\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{\mathbf{a}}_j+{\mathbf{c}}_j}{{\mathbf{b}}_j+{\mathbf{d}}_j}\right]\left({\mathbf{z}}_j\right|\mathbf{\Omega }),\end{array}$$
ⓘ Symbols: $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $g$: positive integer, $h$: positive integer, $\mathbf{\Omega }$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements Permalink: http://dlmf.nist.gov/21.6.E4 Encodings: TeX, pMML, png See also: Annotations for §21.6(i), §21.6 and Ch.21

where ${\mathbf{c}}_j$ and ${\mathbf{d}}_j$ are arbitrary $h$-dimensional vectors. Many identities involving products of theta functions can be established using these formulas.

Let $\mathit{\alpha }$, $\mathit{\beta }$, $\mathit{\gamma }$, $\mathit{\delta }\in {ℝ}^g$. Then

21.6.8		$$\begin{array}{l}\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\gamma }}\right]\left({\mathbf{z}}_1\right|\mathbf{\Omega })\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\beta }}{\mathit{\delta }}\right]\left({\mathbf{z}}_2\right|\mathbf{\Omega })\\ =\sum _{\mathit{\nu }\in {\mathbb{Z}}^g/\left(2{\mathbb{Z}}^g\right)}\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[\mathit{\alpha }+\mathit{\beta }+\mathit{\nu }]}{\mathit{\gamma }+\mathit{\delta }}\right]\left({\mathbf{z}}_1+{\mathbf{z}}_2\right|2\mathbf{\Omega })\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[\mathit{\alpha }-\mathit{\beta }+\mathit{\nu }]}{\mathit{\gamma }-\mathit{\delta }}\right]\left({\mathbf{z}}_1-{\mathbf{z}}_2\right|2\mathbf{\Omega }).\end{array}$$
ⓘ Symbols: $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics, $\in$: element of, $\mathbb{Z}$: set of all integers, $g$: positive integer, $\mathbf{\Omega }$: a Riemann matrix, $\mathit{\alpha }$: $g$-dimensional vector, $\mathit{\beta }$: $g$-dimensional vector, $\mathit{\delta }$: $g$-dimensional vector and $\mathit{\gamma }$: $g$-dimensional vector Permalink: http://dlmf.nist.gov/21.6.E8 Encodings: TeX, pMML, png See also: Annotations for §21.6(ii), §21.6 and Ch.21

Thus $\mathit{\nu }$ is a $g$-dimensional vector whose entries are either $0$ or $1$. For this result and a generalization see Koizumi (1976) and Belokolos et al. (1994, pp. 38–41). For addition formulas for classical theta functions see §20.7(ii).